Polarization-Based Tests of Gravity with the Stochastic Gravitational-Wave Background

The direct observation of gravitational waves with Advanced LIGO and Advanced Virgo offers novel opportunities to test general relativity in strong-field, highly dynamical regimes. One such opportunity is the measurement of gravitational-wave polarizations. While general relativity predicts only two tensor gravitational-wave polarizations, general metric theories of gravity allow for up to four additional vector and scalar modes. The detection of these alternative polarizations would represent a clear violation of general relativity. The LIGO-Virgo detection of the binary black hole merger GW170814 has recently offered the first direct constraints on the polarization of gravitational waves. The current generation of ground-based detectors, however, is limited in its ability to sensitively determine the polarization content of transient gravitational-wave signals. Observation of the stochastic gravitational-wave background, in contrast, offers a means of directly measuring generic gravitational-wave polarizations. The stochastic background, arising from the superposition of many individually unresolvable gravitational-wave signals, may be detectable by Advanced LIGO at design sensitivity. In this paper, we present a Bayesian method with which to detect and characterize the polarization of the stochastic background. We explore prospects for estimating parameters of the background and quantify the limits that Advanced LIGO can place on vector and scalar polarizations in the absence of a detection. Finally, we investigate how the introduction of new terrestrial detectors like Advanced Virgo aid in our ability to detect or constrain alternative polarizations in the stochastic background. We find that, although the addition of Advanced Virgo does not notably improve detection prospects, it may dramatically improve our ability to estimate the parameters of backgrounds of mixed polarization.

Popular Summary

The past two years have seen the first detection of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo Collaborations. Produced by the most energetic and catastrophic events in the Universe—such as collisions of black holes and the explosive deaths of stars—gravitational waves promise to be a valuable test bed for Einstein’s theory of general relativity (GR). So far, GR has withstood all challenges, but it is not the only possible theory of gravity. There are other theories that make different predictions than GR. In the most general theories, gravitational waves can take on six different polarizations, the specific shapes traced by a wave as it moves through space. Interestingly, GR prohibits all but two of these. The observation of any of the other four “forbidden polarizations” would therefore immediately signify new physics beyond the domain of GR. Here, we explore prospects for measuring these polarizations.

The LIGO and Virgo Collaborations have succeeded only very recently in making the first preliminary measurement of a gravitational wave’s polarization, using a short-duration signal (named GW170814) generated by the merger of two black holes. We look to a different class of gravitational-wave source: the stochastic gravitational-wave background, a continuous signal formed by the combination of all gravitational-wave sources that are too distant or too weak to be individually detected. In our work, we present a framework with which to measure the polarization of gravitational waves from the spectral shape of the stochastic background.

We find that current detectors can limit the fraction of the forbidden polarizations in the stochastic background. The addition of future detectors or novel data analysis techniques may improve the ability to distinguish different polarization modes.

Figure 1

Deformation of a ring of freely falling test particles under the six gravitational-wave polarizations allowed in general metric theories of gravity. Each wave is assumed to propagate in the $z$ direction (out of the page for the plus, cross, and breathing modes; to the right for the vector-$x$, vector-$y$, and longitudinal modes). While general relativity allows only for two tensor polarizations (plus and cross), alternate theories allow for two vector ($x$ and $y$) and/or two scalar (breathing and longitudinal) polarization modes.

Figure 2

Overlap reduction functions quantifying the sensitivities of the Hanford-Livingston (left) and Hanford-Virgo (right) baselines to isotropic backgrounds of tensor-, vector-, and scalar-polarized gravitational waves. The distance between Hanford and Virgo is much larger than that between Hanford and Livingston; the Hanford-Virgo overlap reduction functions are therefore smaller in amplitude and more rapidly oscillatory.

Figure 3

Left: PI curves showing the sensitivity of Advanced LIGO to stochastic backgrounds of tensor, vector, and scalar polarizations (solid blue, red, and green curves, respectively). Power-law energy-density spectra [Eq. (10)] drawn tangent to the PI curves have expected $⟨{\mathrm{SNR}}_{\mathrm{OPT}}⟩=3$ after 3 years of observation at design sensitivity. Also shown are “naive” PI curves for vector and scalar backgrounds (dashed red and green curves) illustrating the sensitivity of existing search methods optimized only for tensor polarizations. Right: Minimum detectable background amplitudes ($⟨{\mathrm{SNR}}_{\mathrm{OPT}}⟩=3$ after 3 years of observation at design sensitivity) as a function of spectral index ${\alpha }_a$. For small and negative values of ${\alpha }_a$, Advanced LIGO is approximately equally sensitive to tensor- and vector-polarized backgrounds. For large ${\alpha }_a$, Advanced LIGO is instead most sensitive to vector- and scalar-polarized backgrounds. The dashed curves show amplitudes detectable with existing “naive” methods. The sensitivity loss between optimal and naive cases is negligible for ${\alpha }_a\lesssim 0$, but becomes significant at moderate positive slopes (e.g., ${\alpha }_a\sim 2$). The kinks in the naive curves are due to biases incurred when recovering vector and scalar backgrounds with purely tensor models; see the text for details.

Figure 4

The fractional increase in observing time required for Advanced LIGO to make a detection of vector (red curve) and scalar (green curve) backgrounds using existing search techniques, as a function of their spectral index ${\alpha }_a$. The sharp kinks in each curve are due to biases incurred when fitting vector and scalar backgrounds with a purely tensor model; see the text for details.

Figure 5

Left: Simulated cross-correlation measurements $\widehat{C}(f)$ for purely tensor (blue) and purely scalar (green) stochastic backgrounds, recovered after 3 years of observation with design-sensitivity Advanced LIGO. The backgrounds shown have ${\alpha }_T={\alpha }_S=2/3$, and have amplitudes chosen such that each is detectable with $⟨{\mathrm{SNR}}_{\mathrm{OPT}}⟩=150$. The measured spectra each show distinct modulations characteristic of the tensor and scalar overlap reduction functions, allowing a clear identification of the polarization in each case. Right: Simulated recovery of weaker tensor and scalar backgrounds, detectable with $⟨{\mathrm{SNR}}_{\mathrm{OPT}}⟩=5$ after 3 years of observation at design sensitivity. While each background would be confidently detected by existing search techniques, the characteristic amplitude modulations and hence the polarization content of each simulated background are no longer evident.

Figure 6

Left: Distributions of odds ratios ${\mathcal{O}}_{\mathrm{N}}^{\mathrm{SIG}}$ between signal and noise hypotheses for simulated observations of tensor (blue) and scalar (green) stochastic backgrounds of slope $\alpha =2/3$, assuming 3 years of observation with design-sensitivity Advanced LIGO. We consider two different strengths for each polarization, corresponding to $⟨{\mathrm{SNR}}_{\mathrm{OPT}}⟩=5$ and 10. For each background strength, the tensor and scalar odds ratios lie nearly on top of one another. Also shown is the background distribution of odds ratios obtained when observing pure Gaussian noise (hatched gray). In the presence of a stochastic background, the recovered odds ratios grow as $\ln {\mathcal{O}}_{\mathrm{N}}^{\mathrm{SIG}}\propto {\mathrm{SNR}}_{\mathrm{OPT}}^2$, showing increasingly large preference for the signal hypothesis. Right: Odds ratios ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{NGR}}$ between NGR and GR hypotheses obtained for the same set of simulated Advanced LIGO observations. In the presence of a tensor-polarized background, we recover narrow distributions of odds ratios centered at $\ln {\mathcal{O}}_{\mathrm{GR}}^{\mathrm{NGR}}\approx -1.4$, reflecting consistency with the GR hypothesis. A scalar background, on the other hand, yields large positive odds ratios, correctly showing a strong preference for our NGR hypothesis.

Figure 7

Odds ratios ${\mathcal{O}}_{\mathrm{N}}^{\mathrm{SIG}}$ (left) and ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{NGR}}$ (right) for simulated Advanced LIGO observations of purely tensor- (blue), vector- (red), and scalar-polarized (green) stochastic backgrounds. Within each plot, we show 750 simulated observations, with random log amplitudes chosen uniformly over the range $-10<\log {\mathrm{\Omega }}_0<-7$. Black points mark the results from individual realizations, while the solid curves and shaded regions show the moving mean and standard deviations (smoothed with a Gaussian kernel) of these realizations. For each polarization, $\ln {\mathcal{O}}_{\mathrm{N}}^{\mathrm{SIG}}$ scales quadratically with the amplitude of the stochastic background. Similarly, $\ln {\mathcal{O}}_{\mathrm{GR}}^{\mathrm{NGR}}$ scales quadratically with vector and scalar amplitude. For tensor backgrounds, however, $\ln {\mathcal{O}}_{\mathrm{GR}}^{\mathrm{NGR}}$ instead saturates at approximately $-1.4$.

Figure 8

Odds ratios for simulated Advanced LIGO measurements of stochastic backgrounds containing both tensor and scalar polarizations, assuming 3 years of observation at design sensitivity. The tensor and scalar components have slopes ${\alpha }_T=2/3$ and ${\alpha }_S=0$, respectively. Left: Odds ratios between signal and noise hypotheses. The observed values of ${\mathcal{O}}_{\mathrm{N}}^{\mathrm{SIG}}$ trace contours in total background energy. Thus, the detectability of a background depends largely on its total power, not its polarization content. Right: Odds ratios ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{NGR}}$ between NGR and GR hypotheses. Advanced LIGO would confidently identify the presence of the scalar background component when $\log {\mathrm{\Omega }}_0^S\gtrsim -7.9$. LIGO’s sensitivity to the scalar component is nearly independent of the strength of the tensor component; the minimum identifiable scalar amplitude ${\mathrm{\Omega }}_0^S$ rises only slightly with increasing ${\mathrm{\Omega }}_0^T$.

Figure 9

As above, but for simulated 3-year observations with the joint Advanced LIGO-Virgo network at design sensitivity. Despite the inclusion of Advanced Virgo, the sensitivity of this three-detector network is nearly identical to that of Advanced LIGO alone.

Figure 10

Posteriors obtained for a simulated Advanced LIGO observation of pure Gaussian noise (case 1 in Table 1), under the TVS hypothesis. The subplots along the diagonal show marginalized posteriors for the amplitudes and slopes of the tensor, vector, and scalar backgrounds (blue, red, and green, respectively), while the remaining subplots show the 2D posterior between each pair of parameters. Above each marginalized posterior we quote the median posterior value and $\pm 34\%$ credible uncertainties. Each amplitude posterior is consistent with our lower prior bound, reflecting the nondetection of a stochastic background.

Figure 11

Marginalized amplitude and slope posteriors for the Gaussian noise observation in Fig. 10, after the additional inclusion of design-sensitivity Advanced Virgo. For reference, the light gray histograms show the Advanced-LIGO-only results from Fig. 10. As above, dashed gray lines show the priors placed on each parameter. We see that the inclusion of Advanced Virgo does not significantly affect the parameter estimation results.

Figure 12

As in Fig. 10, for a simulated observation of a pure tensor background (case 2 in Table 1). The injected tensor amplitude and slope are indicated by dot-dashed black lines. The tensor amplitude and slope posteriors are peaked about their true values. The vector and scalar amplitude posteriors, meanwhile, are consistent with our lower prior bound.

Figure 13

Marginalized amplitude and slope posteriors for the tensor background observation in Fig. 12, after the additional inclusion of design-sensitivity Advanced Virgo. For reference, the light gray histograms show the Advanced-LIGO-only results from Fig. 12. The joint LIGO-Virgo parameter estimation yields a slightly tighter measurement of ${\mathrm{\Omega }}_0^T$, as well as somewhat improved upper limits on ${\mathrm{\Omega }}_0^V$ and ${\mathrm{\Omega }}_0^S$.

Figure 14

As in Figs. 10 and 12, for a simulated observation of a mixed tensor and scalar background (case 3 in Table 1). While the ${\mathrm{\Omega }}_0^T$ and ${\mathrm{\Omega }}_0^S$ posteriors are locally peaked about the true values, much of the observed energy is mistaken for vector polarizations. Thus, Advanced LIGO alone is unable to break the degeneracy between tensor, vector, and scalar amplitudes.

Figure 15

Marginalized amplitude and slope posteriors for the mixed tensor and scalar background observation in Fig. 14, after the additional inclusion of design-sensitivity Advanced Virgo. For reference, the light gray histograms show the Advanced-LIGO-only results from Fig. 14. In contrast to the results in Fig. 14, the degeneracy between polarization modes is completely broken when including Advanced Virgo. Thus, while Advanced Virgo does not particularly improve prospects for the detection of a mixed background, it can significantly improve our ability to perform parameter estimation on multiple modes simultaneously.

Figure 16

Odds ratios ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{NGR}}$ obtained for simulated Advanced LIGO observations of tensor-polarized broken power-law backgrounds with energy density spectra given by Eq. (13). The parameters ${\alpha }_1$ and ${\alpha }_2$ are the backgrounds’ slopes below and above the “knee” frequency $f_k$, which we take to be 30 Hz (in the center of the stochastic sensitivity band). We scale the amplitude ${\mathrm{\Omega }}_0$ of each background such that it is optimally detectable with $⟨{\mathrm{SNR}}_{\mathrm{OPT}}⟩=5$ after the simulated observation period. By design, these backgrounds are not well described by single power laws, the form explicitly assumed in our search. Despite this fact, we find that these backgrounds are not systematically misclassified as containing vector or scalar polarization.

Figure 17

Illustration of the prior odds assigned to models and subhypotheses in the hierarchical Bayesian search for generically polarized stochastic backgrounds. When constructing ${\mathcal{O}}_{\mathrm{N}}^{\mathrm{SIG}}$, we assign equal prior probability to the noise and signal models, as well as equal probability to the seven signal subhypotheses $\{\mathrm{T},\dots ,\mathrm{TVS}\}$. Similarly, when constructing ${\mathcal{O}}_{\mathrm{GR}}^{\mathrm{NGR}}$, we give equal probability to the non-GR and GR models and identically weight the six non-GR subhypotheses $\{\mathrm{V},\dots ,\mathrm{TVS}\}$.

Figure 18

MultiNest Bayesian log-evidence for a single simulated stochastic background observation as a function of the number of live points chosen. For the simulated data, we assume a tensor-polarized background (with ${\mathrm{\Omega }}_0^T=2\times {10}^{-8}$ and ${\alpha }_T=2/3$) observed for 1 year with design-sensitivity Advanced LIGO and compute evidence using the T model (see Sec. 4). Results are shown for both MultiNest’s default and INS modes; also shown are the error estimates provided by each mode. To compute the results presented in this paper, we use $n=2000$ live points.

Figure 19

Histograms of MultiNest log-evidence (for the TVS model; see Sec. 4) obtained by evaluating a single simulated data set 500 times in both the default and INS modes. To generate the simulated data, we assume a 1-year observation of a tensor background (${\mathrm{\Omega }}_0^T=2\times {10}^{-8}$ and ${\alpha }_T=2/3$) with design-sensitivity Advanced LIGO. The dashed error bars show the mean 68% confidence interval reported by each method, while the solid error bars show the true 68% confidence interval computed from the log-evidence distributions.